Motor Control, 1998, 2, 101-104 © 1998 Human Kinetics Publishers, Inc.

Is Dynamical Systems Modeling Just Curve Fitting? David A. Rosenbaum The development of mathematical tools for describing dynamical systems has made it possible to characterize forms of behavior that could not be characterized before. This represents progress, but the enterprise runs the risk of being nothing more than curve fitting if investigators fail to identify the physical, biological, or psychological mechanisms which are common to systems that follow the same dynamical regime and which are not common to systems that do not follow the same dynamical regime. Soon after the publication in 1614 of Mirifici Logarithorum Canonis Descriptio (A Description of the Admirable Table of Logarithms), its author, John Napier, was visited by Henry Briggs (1556-163 1), Professor of Geometry at Oxford. "My Lord," Briggs declared, "I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help in astronomy, viz., the logarithms; but, my lord, being by you found out, I wonder nobody found it out before, when now know it is so easy" (Kasner & Newman, 1967, p. 81). Briggs felt compelled to meet Napier because Napier's logarithm made it possible to avoid the tedious calculations that, until then, had to be undertaken for navigation and astronomy. With the logarithm, it later became possible to fit an astonishingly wide range of phenomena. For example, body height is a logarithmic function of age (Palmer, 1944); the perceived brightness of light is a l ogarithmic function of luminance (Gleitman, 1981); the speed of performance on a task grows with the logarithm of practice time on it (Newell & Rosenbloom, 1 981); and the time to move as quickly as possible from one location to another i ncreases with the logarithm of the ratio of twice target distance to target width (Fitts, 1954). That so many phenomena obey a logarithmic relation (and many other phenomena could be added as well) shows that the logarithm is a useful tool for curve fitting. On the other hand, the ubiquity of logarithmic relations needs not imply that these relations have anything more in common than being logarithmic. To say that they do might be as strange as saying that there must be a deep connection between hand tangential velocity profiles (Hogan, 1984) and IQ distributions ( Herrnstein, 1996) because both are bell shaped.

David A. Rosenbaum is with the Department of Psychology, Pennsylvania State University. University Park, PA 16 802. I OI

Whereas Napier's mathematical insights allowed for new computational and statistical advances, the mathematical insights of those who developed dynamical systems-giants like Lorenz and Poincarē-led to a different state of affairs. A number of investigators-many citing the great physicist Maxwell-have sought to identify the same dynamic regime in as many forms as possible. Quoting from a forthcoming chapter on movement coordination, "Characteristic of the dynamical model is its ability to describe coordination with concepts that are indifferent to the particular substrate (e.g., neural, physiological, psychological, computational, social) supporting the movement" (Amazeen, Amazeen, & Turvey, in press). It is indeed exciting to see the same dynamics expressed in different domains. For example, the observation that interactions between the index fingers during bimanual coordination can be captured by the same dynamical equations as the behavior of certain electronic circuits (Kelso, 1984) raises the possibility that these two systems, despite their different material realizations, are the same in some abstract sense. As exciting as such developments are, however, they leave me feeling a bit uneasy. Here I want to raise two questions about dynamical systems which, in spite of the increasing interest in such systems in recent years, have not been raised before, at least to my knowledge. One question is, What characterizes systems that share particular dynamics besides the fact that they share those dynamics? Similarly, what distinguishes systems that do not share particular dynamics besides the fact that their dynamics differ? It is well known that there is a richly varied set of dynamical systems: point attractors, limit-cycle attractors, strange attractors, and so on. Devotees of dynamical systems affectionately refer to the collection of already identified dynamical regimes as a "bestiary." The zoo metaphor is charming, but one wonders why one "beast" applies to one system and a different beast applies to another. The question i s difficult if the systems that have the same dynamics seem to have nothing in common otherwise. For example, congregations of male fireflies in Thailand, Malaysia, and New Guinea flash synchronously to attract female fireflies (Strogatz & Stewart, 1993), much as the two index fingers of participants in studies of bimanual coordination exhibit coupling of the two hands (Kelso, 1984). What is it that these two systems share other than their proclivity for synchronization? I find it hard to answer this question. Understanding the mapping of dynamical regimes to systems becomes even harder when one notes that a system expressing one dynamic and a system expressing a different dynamic may have much in common. For example, though fireflies in Thailand, Malaysia, and New Guinea flash together, fireflies elsewhere do not. Why not? One student of dynamical systems to whom I proposed this question said that only the synchronously flashing fireflies pick up on the relevant dynamic. I wasn't sure if this person was serious or was just, so to speak, winking at me. My other major question about dynamical systems modeling concerns the ontological status of the terms used in the equations. To take an example from Amazeen et al. (in press), these authors reviewed the history of work on bimanual swinging of hand-held pendulums. It turns out that, to account for the motions of these pendulums observed in the incisive series of experiments conducted on this topic, one needs an ever more complex equation, some of whose terms don't

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seem to have any obvious physical or biological referent. I will not rehearse the equation here or point to the specific terms whose ontological status I question. My general point is that having an equation whose terms can be manipulated to provide a fit to data may be quantitatively elegant, especially if, as has been true of the work on bimanual pendulum swinging, the equation grows in a systematic fashion as additional data are accumulated. Still, if the real-world referents for these terms are opaque, questions may be raised about whether the enterprise amounts to sophisticated redescription rather than explanation. Addressing this point depends, of course, on what is meant by explanation. If explanation includes the capacity to make new predictions, then the equations presented by Amazeen et al. have explanatory power because new predictions emerged from them and, in many cases, were confirmed. On the other hand, the fact that the original equation had to be elaborated might be taken as a sign that some of the predictions weren't confirmed and changes in the theory had to be introduced as a result. Another example of a dynamical system model is the Farey tree, which has been used to model the reversion from complex time ratios (e.g., 5:3) to simple ti me ratios (e.g., 1:2) when people try to generate polyrhythms more and more quickly (Peper, Beek, & van Wieringen, 1995). The Farey tree is a beautiful mathematical object based on the Fibonacci series (1, 2, 3, 5, 8, ...). That one can account for changes in produced time ratios via continuous transitions through the Farey tree is intriguing, as is the fact that one can model similar sorts of changes in electrical circuits and chemical systems in the same way (see Peper et al. for references). Still, the abstractness of the model-something many dynamical systems theorists like most about it-leaves me wondering what biological and physical structures and processes the abstract functions correspond to. Without knowing the answer to this question, one returns to the first of my two main questions: Why do some dynamical systems apply to certain systems but not others? Why, for example, don't we hop as we run faster and faster, just as we tend to fall into an in-phase mode of bimanual finger oscillation when we swing our fingers more and more quickly-a question I have raised before (Rosenbaum, 1991, chapter 11)? Clearly, factors distinguishing locomotion from finger swinging set these two tasks apart. If the dynamical systems approach cannot identify these factors, then one begins to wonder about the explanatory power of the approach. All these considerations bring me back to the question that serves as the title for this commentary: Is dynamical systems modeling just curve fitting? I would say, Yes it is. But when I answer in the affirmative, I am actually playing on words, for the word just can mean "valid" or "worthwhile" as well as "mere" or "trivial." When I say that dynamical systems modeling is just curve fitting, I actually mean that it is a warranted form of quantitative description. In the same way that Napier's l ogarithm made it possible to describe a whole range of phenomena that would have been hard to describe otherwise, the nonlinear equations used in dynamical systems modeling provide a new way to characterize patterns whose complexity would resist description otherwise. Nonetheless, as I have suggested here, the approach also raises many questions. If we can learn why some dynamics characteri ze some systems but not others and therefore what, at a deeper level, accounts for the organization of the dynamical bestiary, we may go much farther in understandi ng motor control and nature more generally.

Rosenbaum

References Amazeen, P.G., Amazeen, E.L., & Turvey, M.T. (in press). Dynamics of human intersegmental coordination: Theory and research. In D.A. Rosenbaum & C. Collyer (Eds.), Timing of behavior: Neural, psychological, and computational perspectives. Cambridge, MA: MIT Press. Fitts, P.M. (1954). The information capacity of the human motor system in controlling the amplitude of movement. Journal of Experimental Psychology, 47, 381-391. Gleitman, H.G. (1981). Psychology. New York: Norton. Herrnstein, R.J. (1996). The bell curve: Intelligence and class structure in American life. New York: Free Press. Hogan, N. (1984). An organizing principle for a class of voluntary movements. Journal of Neuroscience, 4, 2745-2754. Kasner, E., & Newman, J. (1967). Mathematics and the imagination. New York: Simon & Schuster. Kelso, J.A.S. (1984). Phase transitions and critical behavior in human bimanual coordination. American Journal of Physiology: Regulatory, Integrative and Comparative, 246, R1000-R1004. Newell, A.M., & Rosenbloom, P.S. (1981). Mechanisms of skill acquisition and the law of practice. In J.R. Anderson (Ed.), Cognitive skills and their acquisition (pp. 1-55). Hillsdale, NJ: Erlbaum. Palmer, C.E. (1944). Studies of the center of gravity in the human body. Child Development, 15, 99-163. Peper, C.E., Beek, P.J., & van Wieringen, P.C.W. (1995). Multifrequency coordination in bimanual tapping: Asymmetrical coupling and signs of supercriticality. Journal of Experimental Psychology: Human Perception and Performance, 21, 1117-1138. Rosenbaum, D.A. (1991). Human motor control. San Diego, CA: Academic Press. Strogatz, S.H., & Stewart, I. (1993, December). Coupled oscillators and biological synchronization. Scientific American, pp. 102-109.

Author Notes Preparation of this article was aided by Grant SBR-94-96290 from the National Science Foundation, a grant from the Research and Graduate Studies Office of Penn State University, and Research Scientist Development Award K02-MH00977-O1A1 from the National Institute of Mental Health. Accepted for publication: November 3, 1997